Abstract
One of the problems in Computer Vision is recovery of object shapes from noisy images. Associated with this problem is the question of what is a shape and how is it to be represented. Since answers to these questions have to be ultimately tailored to the uses one has in mind, one has to bring into consideration potential applications and with it, the question of practical algorithms for implementation of the theory. Here we are concerned with mainly two-dimensional shapes. Mathematically, an object is simply an open subset in the image domain, characterized in some way. In the real world, what is an object and what is just noise or clutter depends of course on what one is looking for. For example, Figure la shows a noiseless synthetic image. It may be reasonable to assume that the objects in the figure are the four squares in the four corners and the two ellipses in the middle. Figure lb shows a noisy version obtained from the image in Figure la by adding Gaussian noise. The signal-to-noise ratio (i.e. the ratio between the standard deviation of the image with noise removed and the standard deviation of the noise) is 1: 4. The problem is to recover the original objects.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This research was partially supported under ARO Grant, No. DAAL03–91-G-0041.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
Bibliography
L. Alvarez, P.L. Lions and J.M. Morel: “Image Selective Smoothing and Edge Detection by Nonlinear Diffusion II”, SIAM J. Num. Anal., (June, 1992).
L. Ambrosio and V.M. Tortorelli: “Approximation of Functionals Depending on Jumps by Elliptic Functionals via r-convergence”, Arch. Rat. Mech. Anal. 111, pp. 291–322, (1990).
L. Ambrosio and V.M. Tortorelli: “On the Approximation of Functionals Depending on Jumps by Quadratic, Elliptic Functionals”, Boll. Un. Mat. Ital. (1992).
G. Bellettini, G. Dal Maso and M. Paolini: “Semicontinuity and Relaxation Properties of a Curvature Depending Functional in 2d”, SISSA Report, 17/92/MA, (Feb. 1992).
A. Blake and A. Zisserman: “Using weak continuity constraints”, Report CSR-186–85, Dept. of Comp. Sci., Edinburgh Univ. (1985).
A. Blake and A. Zisserman: Visual Reconstruction, M.I.T. Press (1987).
G. Dal Maso, J.M. Morel and S. Solimini: “A Variational Method in Image Segmentation: Existence and Approximation Results”, Acta Math. 168, pp. 89–151, (1989).
E. De Giorgi, M. Carriero and A. Leaci: “Existence Theorem for a Minimum Problem, with Free Discontinuity Set”, Arch. Rat. Mech. Anal. 108, pp. 195–218, (1990).
S. Geman and D. Geman: “Stochastic relaxation, Gibbs’ distributions, and the Bayesian restoration of images”, IEEE Trans., PAMI 6, pp. 721–741, (1984).
M. Kass, A. Witkin and D. Terzopoulos: “Snakes: Active Contour Models”, First International Conf. on Computer Vision, (1987).
D. Mumford: “Elastica and Computer Vision”, Preprint.
D. Mumford and J. Shah: “Boundary detection by minimizing functionals, I”, Proc. IEEE Conf. on Computer Vision and Pattern Recognition, San Francisco, (1985).
D. Mumford and J. Shah: “Optimal approximations by piecewise smooth functions and associated variational problems”, Comm. on Pure and Appl. Math., v. XLII, n. 5, pp. 577–684 (July, 1989).
M. Nitzberg and D. Mumford: “The 2.1D Sketch”, Third Int’l Conf. on Comp. Vision, (December, 1990).
S. Osher and J. Sethian: “Fronts Propagating with Curvature Dependent Speed: Algorithms based on the Hamilton-Jacobi Formulation”, J. Comp. Physics, 79, (1988).
M. Proesmans, E.J. Pauwels, L.J. Van Gool and A. Oosterlink: “Image Enhancement using Non-Linear Diffusion”, Proc. IEEE Conf. on Comp. Vision and Pattern Recognition, (June, 1993).
M. Proesmans, L.J. Van Gool and A. Oosterlink: “Determination of Optical Flow and its Discontinuities using Non-linear Diffusion”, Preprint, ESAT-MI2, Katholike Univ. Leuven, Leuven, Belgium, (1993).
T. Richardson: Ph.D. Thesis, Department of Electrical Engineering and Computer Science, MIT (1990).
J. Shah: “Parameter Estimation, Multiscale Representation and Algorithms for Energy-Minimizing Segmentations”, Tenth International Conference on Pattern Recognition, (June, 1990).
J. Shah: “Segmentation by Nonlinear Diffusion”, Proc. IEEE Conf. on Comp. Vision and Pattern Recognition, (June, 1991).
J. Shah: “Segmentation by Nonlinear Diffusion, II”, Proc. IEEE Conf. on Comp. Vision and Pattern Recognition, (June, 1992).
J. Shah: “Properties of Energy-Minimizing Segmentations”, SIAN J. on Control and Optim. v. 30, no. 1, 99–111, (1992).
J. Shah: “A Nonlinear Diffusion Model for Discontinuous Disparity and Half-Occlusions in Stereo”, Proc. IEEE Conf. on Comp. Vision and Pattern Recognition, (June, 1993).
J. Shah: “Piecewise Smooth Approximations of Functions”, Calculus of Variations and Partial Differential Equations, 2, pp. 315–328, (1994).
J. Shah: “Shape Recovery from Noisy Images by Curve Evolution”, IASTED International Conference on Signal and Image Processing, (Nov. 1995).
J. Shah: “A Common Framework for Curve Evolution, Segmentation and Anisotropic Diffusion”, Proc. IEEE Conf. on Computer Vision and Pattern Recognition, San Francisco (1996).
S. Tari, J. Shah and H. Pien: “A Computationally Efficient Shape Analysis via Level Sets”, IEEE Workshop on Mathematical Methods in Biomedical Image Analysis (1996).
Y. Wang: Ph.D. Thesies, Dept. of Math., Harvard Univ., (1989).
Y. Wen: “L 2 Flow of Curve Straightening in the Plane”, Duke Math. J. v. 70, n. 3, pp. 683–398, (June, 1993).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1996 Birkhäuser Verlag Basel/Switzerland
About this paper
Cite this paper
Shah, J. (1996). Uses of Elliptic Approximations in Computer Vision. In: Serapioni, R., Tomarelli, F. (eds) Variational Methods for Discontinuous Structures. Progress in Nonlinear Differential Equations and Their Applications, vol 25. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9244-5_3
Download citation
DOI: https://doi.org/10.1007/978-3-0348-9244-5_3
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-0348-9959-8
Online ISBN: 978-3-0348-9244-5
eBook Packages: Springer Book Archive